3.35/1.65 YES 3.35/1.66 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.35/1.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.35/1.66 3.35/1.66 3.35/1.66 Quasi decreasingness of the given CTRS could be proven: 3.35/1.66 3.35/1.66 (0) CTRS 3.35/1.66 (1) CTRSToQTRSProof [SOUND, 0 ms] 3.35/1.66 (2) QTRS 3.35/1.66 (3) QTRSRRRProof [EQUIVALENT, 53 ms] 3.35/1.66 (4) QTRS 3.35/1.66 (5) QTRSRRRProof [EQUIVALENT, 0 ms] 3.35/1.66 (6) QTRS 3.35/1.66 (7) RisEmptyProof [EQUIVALENT, 0 ms] 3.35/1.66 (8) YES 3.35/1.66 3.35/1.66 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (0) 3.35/1.66 Obligation: 3.35/1.66 Conditional term rewrite system: 3.35/1.66 The TRS R consists of the following rules: 3.35/1.66 3.35/1.66 a -> c 3.35/1.66 a -> d 3.35/1.66 3.35/1.66 The conditional TRS C consists of the following conditional rules: 3.35/1.66 3.35/1.66 f(x, y) -> z <= c -> x, y -> z 3.35/1.66 3.35/1.66 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (1) CTRSToQTRSProof (SOUND) 3.35/1.66 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (2) 3.35/1.66 Obligation: 3.35/1.66 Q restricted rewrite system: 3.35/1.66 The TRS R consists of the following rules: 3.35/1.66 3.35/1.66 f(x, y) -> U1(c, y) 3.35/1.66 U1(x, y) -> U2(y) 3.35/1.66 U2(z) -> z 3.35/1.66 a -> c 3.35/1.66 a -> d 3.35/1.66 3.35/1.66 Q is empty. 3.35/1.66 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (3) QTRSRRRProof (EQUIVALENT) 3.35/1.66 Used ordering: 3.35/1.66 Polynomial interpretation [POLO]: 3.35/1.66 3.35/1.66 POL(U1(x_1, x_2)) = 2*x_1 + 2*x_2 3.35/1.66 POL(U2(x_1)) = 2*x_1 3.35/1.66 POL(a) = 1 3.35/1.66 POL(c) = 1 3.35/1.66 POL(d) = 0 3.35/1.66 POL(f(x_1, x_2)) = 2 + x_1 + 2*x_2 3.35/1.66 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.35/1.66 3.35/1.66 a -> d 3.35/1.66 3.35/1.66 3.35/1.66 3.35/1.66 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (4) 3.35/1.66 Obligation: 3.35/1.66 Q restricted rewrite system: 3.35/1.66 The TRS R consists of the following rules: 3.35/1.66 3.35/1.66 f(x, y) -> U1(c, y) 3.35/1.66 U1(x, y) -> U2(y) 3.35/1.66 U2(z) -> z 3.35/1.66 a -> c 3.35/1.66 3.35/1.66 Q is empty. 3.35/1.66 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (5) QTRSRRRProof (EQUIVALENT) 3.35/1.66 Used ordering: 3.35/1.66 Knuth-Bendix order [KBO] with precedence:a > f_2 > U1_2 > U2_1 > c 3.35/1.66 3.35/1.66 and weight map: 3.35/1.66 3.35/1.66 c=1 3.35/1.66 a=1 3.35/1.66 U2_1=1 3.35/1.66 f_2=0 3.35/1.66 U1_2=0 3.35/1.66 3.35/1.66 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.35/1.66 3.35/1.66 f(x, y) -> U1(c, y) 3.35/1.66 U1(x, y) -> U2(y) 3.35/1.66 U2(z) -> z 3.35/1.66 a -> c 3.35/1.66 3.35/1.66 3.35/1.66 3.35/1.66 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (6) 3.35/1.66 Obligation: 3.35/1.66 Q restricted rewrite system: 3.35/1.66 R is empty. 3.35/1.66 Q is empty. 3.35/1.66 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (7) RisEmptyProof (EQUIVALENT) 3.35/1.66 The TRS R is empty. Hence, termination is trivially proven. 3.35/1.66 ---------------------------------------- 3.35/1.66 3.35/1.66 (8) 3.35/1.66 YES 3.35/1.69 EOF